Of tortoises and school buses

This article is for the students, especially those who are enjoying a peaceful life after the board exams. If you belong to this lucky group, you might want to spend a little time reading this bit of wisdom from the past. It’s actually like a cool FB post, only about 2500 years old, by a guy called Zeno. Make friendship with Zeno now and he might help you in college, with all the math’s stuff you are going to find there.

2500 years ago, around the same time Buddha was preaching in India, the little country of Greece was teeming with all kinds of scholars. They had no FB and WhatsApp to share their status updates, but their teaching was earnestly learned and kept alive by their students. One such scholar was Zeno.

Zeno liked giving strange little problems to others. He told people to imagine a race – a race between a tortoise and a young man called Achilles. I mean, isn’t this like your school cricket team against Australia? Everyone knows how this race will end. But Zeno gave them a different story. It goes like this:

Suppose Achilles is ten times faster than the tortoise. To make the race fair, let’s place the tortoise 100 meters ahead of Achilles.

Let the race begin.

First, Achilles has to run the 100 meters. The tortoise is ten times slower. So it will run 10 meters while Achilles covers the 100 meters gap. Now there are 10 meters between them.

Next, Achilles has to run 10 meters. In this time, our tortoise is going to run 1 meter.

Achilles does not take long to run 1 meter, but tortoise has raced ahead by 10 centimetres!

Take out a forgotten notebook now, and try to write the next steps in this story. Well, you will finish all the notebooks, but this story will never end. No matter what you do, the tortoise will remain a little bit ahead. Achilles will never overtake the tortoise.

But we know Achilles actually overtakes the tortoise in real life. Such things are called Paradoxes. A Paradox is something you know to be false, but you can’t point out what is wrong with it.

There is another way in which Zeno’s Paradox is told. Suppose your school bus is standing 100 meters from you. You have to run and catch it. Now, to cover the 100 meters you have to first run 50 meters. Now only 50 meters remain. But before you run 50 meters, you have to go 25 meters. You run half the distance, but half still remains. No matter how much you run, a tiny bit will still remain. This means you can never catch the bus, even though it is standing still! But in real, you do catch the bus, almost everyday. That's what a Paradox is.

When scholars see such Paradoxes, their eyes light up. ‘Let me try my hand at this now’, they say to themselves. And so over the years, many clever people have tried to answer Zeno in their own way. Some fellows have applied mathematical ideas to the problem. You will probably like one of these ideas - it’s called a ‘convergent series’.

To understand convergent series, let’s go to the school bus again and try to cover the 100 meters. The distance you cover first is 50, then 25 and so on. Let me write it down:

50, 25, 12.5, 6.25, 3.125, and many more. To get the next number, just make half of the last one.

Let’s add all these meters that you ran:

50 + 25 + 12.5 + 6.25 + 3.125 + ...

(These three little dots at the end mean -‘Picture abhi baki hai’, or the story does not end here, there are going to be many such numbers).

What do you think the total of all these numbers will be?

Remember that you were 100 meters away from the bus? So the total will never be more than 100. But it will never be exactly 100 too. See for yourself:

50
75
87.5
93.75
96.875
...

Can you calculate the next totals?

You are going to meet these funny convergent series in college. The method to find totals of such series is called Integration, and once you master integration and things like that, there is no stopping you. One day you will possibly build a Jupiter Mission for India.